Question: Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{q^3 - 15q^2 + 50q}{q^3 + 5q^2 - 50q}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {q(q^2 - 15q + 50)} {q(q^2 + 5q - 50)} $ $ x = \dfrac{q}{q} \cdot \dfrac{q^2 - 15q + 50}{q^2 + 5q - 50} $ Simplify: $ x = \dfrac{q^2 - 15q + 50}{q^2 + 5q - 50}$ Since we are dividing by $q$ , we must remember that $q \neq 0$ Next factor the numerator and denominator. $ x = \dfrac{(q - 5)(q - 10)}{(q - 5)(q + 10)}$ Assuming $q \neq 5$ , we can cancel the $q - 5$ $ x = \dfrac{q - 10}{q + 10}$ Therefore: $ x = \dfrac{ q - 10 }{ q + 10 }$, $q \neq 5$, $q \neq 0$